Uniform Bessel bounds for magnetization crossings in finite Ising chains
Abstract
We consider a finite Ising chain in zero field and ask when the all-up configuration is as likely as zero magnetization (magnetization one for an odd number of spins). For both free and periodic ends, the crossing coupling of a long chain is half the logarithm of its length minus half the logarithm of that logarithm, up to a term that vanishes as the length grows. We compute further terms with explicit constants and, for free ends, bound the crossing at every length. The second term is missing from an earlier approximation and explains why an earlier numerical fit gives a slope below one half. To locate the crossing, we compare the known magnetization law with its Bessel approximation. The Bessel approximation is above the exact weight, and we bound its relative error for every length, coupling and magnetization. At the crossing this bound tends to zero. The central bin still has about as many domain walls as the logarithm of the length on average, so the crossing has a coupling width of order the inverse logarithm. In a fixed positive dimensionless field the last crossing comes from near the edge of the row, at a coupling of about a quarter of the logarithm of the length. For long chains it is the one-spin crossing when the field exceeds half the logarithm of two. Below that value another bin crosses later.
Related
- Uniform lower bounds for the average length of finite Pierce expansions
- Target-adapted Green-Bessel SVGD: uniform-in-time propagation of chaos and last-iterate consistency
- Optimized bounds for the product and the ratios of modified Bessel functions
- Sharp two-sided bounds for the product of modified Bessel functions and for its logarithmic derivative
- Finite element approximation for the Bessel (p,s)-Laplacian
BibTeX
@misc{pemmasani2026uniform,
title = {{Uniform Bessel bounds for magnetization crossings in finite Ising chains}},
author = {Arjun Pemmasani},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06935},
eprint = {2610.06935},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06935},
}
RIS
TY - UNPB
TI - Uniform Bessel bounds for magnetization crossings in finite Ising chains
AU - Arjun Pemmasani
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.06935
UR - https://arxiv.org/abs/2610.06935
AB - We consider a finite Ising chain in zero field and ask when the all-up configuration is as likely as zero magnetization (magnetization one for an odd number of spins). For both free and periodic ends, the crossing coupling of a long chain is half the logarithm of its length minus half the logarithm of that logarithm, up to a term that vanishes as the length grows. We compute further terms with explicit constants and, for free ends, bound the crossing at every length. The second term is missing from an earlier approximation and explains why an earlier numerical fit gives a slope below one half. To locate the crossing, we compare the known magnetization law with its Bessel approximation. The Bessel approximation is above the exact weight, and we bound its relative error for every length, coupling and magnetization. At the crossing this bound tends to zero. The central bin still has about as many domain walls as the logarithm of the length on average, so the crossing has a coupling width of order the inverse logarithm. In a fixed positive dimensionless field the last crossing comes from near the edge of the row, at a coupling of about a quarter of the logarithm of the length. For long chains it is the one-spin crossing when the field exceeds half the logarithm of two. Below that value another bin crosses later.
ER -